To get there by public transport, take the subway to Yeonshinnae (연신내) station, exit 3, and from there take bus 34 or 704 to 북한산성입구 (Bukhansan fortress). WARNING: This journey is HELL. There are so many hikers packing onto every bus that comes, you are guaranteed to be crushed up against an old man for the whole journey, trying not to fall over every time the bus goes around a corner. It really is the worst bus journey ever, and sometimes you have to wait ages at the bus stop. Luckily it doesn't last TOO long. From there you'll see the entrance to Bukhansan, a wide road flanked by outdoor clothing brand stores. Follow the signs towards 백운대.
Have a look over at my other blog for some photos of the beautiful mountain scenery, and check out our more recent Bukhansan hiking video on Youtube: